I have rectangle with co-ordinates(x1,y1) and (x2,y2) and I have to rotate the rectangle an amount of θ about it centre using Rotation Matrix

```
| cosθ sinθ |
| -sinθ cosθ |
```

I need to find the co-ordinates of bounding rectangle after rotation.

**Before rotation**

```
0,0
|"""""""""""""""""""""""""""""""""""""""""""|
| |
| x1,y1 |
| |"""""""""""""| |
| | | |
| | | |
| | | |
| """""""""""""" x2,y2 |
| |
| |
""""""""""""""""""""""""""""""""""""""""""" W,H
```

**After rotation**

```
0,0
|"""""""""""""""""""""""""""""""""""""""""""|
| ?,? |
| |""""/\"""""| |
| | / \ | |
| | / \ | |
| | / /| |
| |/ / | |
| |\ / | |
| | \ / | |
| | \ / | |
| """"""""""" ?,? |
| |
| |
""""""""""""""""""""""""""""""""""""""""""" W,H
```

Is there any general equation for finding the co-ordinates of bounding rectangle?.

Thanks….

Haris.

## Best Answer

For each of the four original corners: $$ (x_1,y_1), (x_1,y_2), (x_2,y_1), (x_2,y_2) $$ use the rotation matrix to obtain four new corners. Then to obtain the two ordered pairs that define your bounding rectangle, let one ordered pair have the minimum $x$ and $y$ values out of all the new corners and let the other ordered pair have the maximum $x$ and $y$ values.